A plane has $360$ total seats, which are divided into economy class and business class. For every $13$ seats in economy class, there are $5$ seats in business class. How many seats are there in each class? There are
Explanation: Let $x$ represent the number of seats in the economy class and let $y$ represent the number of seats in the business class. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that the plane has $\textit {360}$ seats divided into the economy class and the business class. How can we model this sentence algebraically? Since the number of seats in the economy class and the business class add up to $360$ seats, we get the following equation: $x+ y = 360$ We are also given that for every $\textit{13}$ seats in the economy class, there are $\textit{5}$ seats in the business class. This can be expressed as: $\dfrac{x}{13}=\dfrac{y}{5}$ Let's rewrite this equation so that it's solved for $x$ : $x = \dfrac{13y}{5}$ Now that we have a system of two equations, we can go ahead and solve it! Let's substitute $ x={ \dfrac{13y}{5}}$ into the first equation: $\begin{aligned} x+ y &= 360\\\\ \left( { \dfrac{13y}{5}}\right)+y&=360\\\\ \dfrac{13y+5y}{5} &=360\\\\ 18y&=360 \cdot 5\\\\ y&=100\end{aligned}$ Now we can substitute $y = 100$ into $x+y=360$ and find that $x=260$. Recall that $x$ denotes the number of seats in the economy class and $y$ denotes the number of seats in the business class. Therefore, there are $\textit{260}$ seats in the economy class and $\textit{100}$ seats in the business class.